NBS TCPUMIPAL NOTE

430

Designs for Surveillance of the Volt Maintained By a Small Group of Saturated Standard Cells

W. G. Eicke and J. M. Cameron

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards
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UNITED STATES DEPARTMENT OF COMMERCE

Alexander B. Trowbridge, Secretary NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director ^J^- TECHNICAL NOTE 430

ISSUED OCTOBER 9, 1967

Designs for Surveillance of the Volt Maintained by a Small Group of Saturated Standard Cells

W. G. Eicke

Electricity Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

J. M. Cameron

Applied Mathematics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office Washington, D.C, 20402 - Price 20 cents
When a local standard such as that for electromotive force Is maintained by a group of

standards, procedures must be established to provide evidence that the group has maintained its original value. One also needs methods for the transfer of the value to test items that provide

efficient use of measurement effort while monitoring the measurement process and- providing

information for updating the values of process parameters. Solutions to the more general problem

of transferring the value from laboratory to laboratory and of maintaining agreement among labora- tories depend on the existence of control within the laboratories. This note is one of a number of contemplated reports having the general aim of providing

methods for the surveillance of measurement processes with emphasis on the amount and kind of infor- mation needed for the estimation and control of the uncertainty in measurement.

August 1967 M. B. Wallenstein Acting Director, Institute for Basic Standard s
W. G. Eicke Electricity Division

Electrochemi stry Sectior

J. M. Cameron Applied Mathematics Division Statistical Engineering Laboratory

This technical note describes a procedure for maintaining surveillance crv small group of saturated standard cells. The measurement process is briefly

discussed and the principle of left-right balance as a means of eliminating eg systematic errors is developed. Specific designs and their analysis for intei comparing 3, ^, 5 and 6 cells in a single temperature controlled environment e given. Procedures for setting up control charts on the appropriate parameters

given, and a technique is described for detecting certain types of systematic

I. INTRODUCTION

At the local level the primary standard of electromotive force is maintained by a group of saturated standard cells, the same type of cell used to maintain the National unit of electromotive force. Many laboratories use groups containing from 3 to 6 cells mounted in either a tem- perature controlled air or oil bath. The cells are in general calibrated by the National Bureau

of Standards at periodic intervals and the mean emf of the group is assumed to remain constant between calibrations. Since such calibrations are done infrequently (at intervals of one year

or more) some technique must be employed to maintain surveillance over the local unit between

calibrations. Starting with assigned values for each of the cells of the group as a given set of reference

points, one can check on the relative stability of the cells by measuring differences among them. One could measure all possible differences and have equal precision in the knowledge of the values

of all cells or one could pick a favorite cell and compare all others with this one (but this leads

to high precision in the knowledge of the selected cell and relatively low precision in all

others). For small groups it is quite practical to measure all possible differences, but as the group size increases the number of measurements would increase rapidly with K, where N is the group

size. As N increases compromise schemes that lead to equal precision in the knowledge of each cell

can be used. One such design is given in [l] for a group of 20 cells in which only 40 differences

are measured (instead of 190 if all N(N-l)/2 were measured). This note discusses methods for maintaining surveillance of groups containing three, four,

five, or six cells in a single temperature controlled enclosure. The procedures suggested are designed to yield information on:

1) the stability of the differences in emf among the group,

2) the components of variability and dependence of the measurement process precision on environmental influences or procedural changes,

3) possible systematic errors and estimation of the accuracy of the process. Furthermore, they tend to maximize the yield of useful information per measurement.
II. The Measurement Technique

The opposition method [3] is usually employed in the intercomparison of saturated standard

cells. In this method the small difference between two cells connected in series opposition is measured using a suitable instrument. The instrument is usually a potentiometer designed for the measurement of very small emfs [2]

.

In the ideal situation the difference in emf as measured by the potentiometer is: AE = Vt - Va (1) where V! and V2 are the emfs of the two cells being compared. However in the real situation there may be spurious emfs in the circuit. In general these can

be classified into two categories;

/ely so, in relation to the interval

d the interval over which a complete

:f the emfs are of the second type they will have the effect of decreasing the precision of the

irocess. On the other hand if they are of the first type they will have the effect of introducing

. systematic error, thereby making the tr AE = Vj - V2 + P where P is the c onstant emf. It is possible to es :imate P by taking

AE' = V2

- Vi + P and summing the two

(2)

(3) w

(5) said to be "left final result.

is in precision

from the operLativ - to the

2P = AE + AE' The difference between eqs. (2) and (3) gives

AE - AE' = 2(Vi - V2 ) an estimate of Vi - V3 free of P. The pair of measurements (eqs. (2) and (3)) < right" balanced. That is if there is a positional effect it is balanced out of t This technique is analogous to that used to eliminate the inequality of balance s weighing on a two pan equal arm balance. In order to designate the cell positi< ational point of view they are frequently designated as unknown and reference:

input terminals of the measuring instrument they are as shown in fig. 1. In the next section the principle of "left-right" balance will be extended to groups -containing three or more cells.

III. Designs for Groups of 3, h, 5, or 6 Cells Experimental designs of groups of 3, 4, 5, and 6 cells are given in Appendix A. The designs presented have been selected to be (l) efficient from the standpoint of the operators making the measurement (2) statistically efficient, in the sense of minimum standard deviation for the esti- mated cell values, and (3) relatively easy to analyze using conventional desk calculators. All of the analyses presented are the least square solutions for the associated design assuming that the

sum of the differences from the mean of all is zero. For groups of 3, h, 5, total "left-right" balance has been achieved and the estimate of the "left-right" effect is

where n is the number of measurements and y the observed difference in emf between two cells.*

the case of six cells left-right balance is achieved for only the first 12 measurements.

In terms of the notation of fig. 1 y is the observed AE.

4
+

T '

)

Ex—E _#

(UNK) (REF) AE = Ex -E R +Ep Negative terminals of cells connected together Measuring Instrument j_4

<-AE-»

(unk) Cref) AE s Ex - ER +-Ep

Positive terminals of ceils connected together

Two ways in which two cells can be connected in series-opposition.

-3-
For each size group the design and its analysis are given as a complete entity, with the left half giving the general procedure and the right half a numerical example. The suggested order for making the measurements requires moving one set of leads at a time thereby minimizing the possibility of connecting the wrong cells. The definitions of the symbols used in the tables are as follows: Symbol Definition

5f. The emf of the ith cell M The group mean

M* The mean of

v. The difference V

± -H

y. The ith measured difference

The calculated circuit residual

Q,± A sum of those y's that involve the ith cell y

± The calculated ith v

y. The predicted y. calculated from v and P

d. J Deviations, (y. - y .

)

s .-- The standard deviation of a single observation- For all of the designs given in Appendix A the assumption is that the mean of the whole group

is known and serves. 'as the restraint in the least square solution'. In the next section, procedures

for changing the- restraint will be given. The analysis produces the following basic information

which can be use'd to monitor the process:

1. The emf of each cell (or the difference from the group mean)

2. The residual eV^fj' P

. 3. The standard deviation of a single observation

h. The deviation of each observation from the predicted value

The frequency with which these intercomparisons should be ran may vary considerably depending

on the particular installation. Once it is established that the process is in a state of control then one intercomparison each week should be sufficient.

' - . :..'•, IV. Change of Restraint

In the previous section it was assumed that the mean for the whole group was known, such as would be -the case if the group had been assigned values by the National Bureau of Standards. Because only differences in emf are measured, this average value is the restraint on the values which provide the "ground zero" to which the cell values are related. When one or more of the cells show a change so large as to be inconsistent with its assigned

- value, it becomes necessary to remove these cells from- the -defining group. Evidence of such changes

'would be discovered 'from control charts on either the. cell values or control charts on differences between cells (see the next .section on control charts).

To illustrate, let -us assume that the- assigned values for the 5th and 6th cells of the example

"in Table A-^ had"' been 1.0182 536 and 1.0182501 instead- of the values (l.0l824l6, 1 .0182381) given in
section 1 of the table. This is a change of +12.0/JV i the cells would "be as shown in the following table.

? emf values for

Table 1-A (Table A-4

Sec. 4) Emf of cell Difference Assigned values

10.47 M + Vi from assigned 1.0182605 1.01826445 3-95

2655 15.62 I.OI82696O 4.10

2466 - 3.4b 1.01825058 3.98 2^76 - 2.29 1.01825169 4.09

2536* - 8.37 1.01824561 -7.99 2501* -12.04 1.01824194 -8.16

L. 01825398 = M

values differ by 12j*V from t ; data of Table A-4.

The last 1 want to remove

;lls are obviously inconsistent with their a

: from the restraint and establish "ground z signed values, so that one wouldro" with the first four cells.

: the average of the assigned value of the c

> shown in column 2 of the table below,-

i retained in the rest

(2) v.: the cell est

(3) adds V to each c

1 section 4 and copied into column 3 below; and

.he cell values, V., as shown in column 4.

(Table A-4

Sec. 4)

1 1.0182605

2 2655

3 2466

4 2476

Average = 1.01825505 = vA

5 1.0182536

6 2501

10.47 15.62

- 3.40

- 2.29

- 8.37

-12.04

1.01826042

26557 24655 24766

24158 23792

The cell values are now expressed in

point. The misbehaving cells would ordins stabilize at some new value.

erms of the average of the "good" cells as the reference

ily continue to be measured in the hope that they would
V. Control Charts Control charts [5] [6] on process parameters such as the cell values and standard deviations

of a single observation provide an effective means of determining whether or not the process is in

a state of statistical control. Control charts for each cell (or difference between cells), process precision (standard deviation of an observation), and the residual emf P should be maintained. These charts provide the verification of that part of the uncertainty statement that deals with

bounds for the effect of random error. Such statements say in effect "If this measurement process

is used a large number of times, the values obtained for a single quantity will vary within the

stated limits." The charts permit one to demonstrate the validity of such statements on current

For each run one will have values for each of the cells, the standard deviation, and the residual emf. To check on the state of control of the measurement process and on the stability of the cells, one would study the sequence of values for these parameters. Control charts on the cells can be established on the emf of the cells, the difference between successive cells (e.g. cell 1 - cell 2, cell 2 - cell 3, cell 3 - cell k, etc.), or both. The former has the difficulty that it is not sensitive to a change in the emf of a single cell. However, by following the differences between successive cells (i.e. 1st minus 2nd, 2nd minus 3rd and so on) one has an easily inter- preted set of results even though the successive differences are not independent. A single "bad" cell will show up as out of control on two successive differences, whereas the remaining differences

are unaffected.

In order to establish control limits one has to know the precision of the measurement process

(see discussion on measurement processes in ref. [7]). However, under the assumption that the

standard deviation, O, of the process is known one can, for a given design, write down the

standard deviations of the individual cell emfs, the difference between two cells, and for the residual P. One can use three times the appropriate standard deviation as control limits. For the designs given the values for setting limits are shown in Table 2. Unfortunately, when starting such surveillance the process precision, a, is usually not known

and must be estimated from the available data. In this case one would pool a number, m, of individual standard deviations using the formula 4 = 1/m 2 S? (7) i=l

for a particular design. This might entail making several runs a week for the first month or so to obtain starting control limits. After about 100 degrees of freedom have been accumulated a new

value of Sj, should be calculated and the control limits revised. For such a large number of degrees of freedom sp approaches a very closely. Using sp and the control limits from the factors

in Table 2, a control chart on the standard deviation of the process (i.e. on s as computed in Appendix A) should be constructed.

It is also desirable to maintain a control chart on the residual emf using the limits given in Table 2. Initially the accepted value of P would be taken as zero. However, if after repeated measurements the value of P is other than zero and constant, the central value and control limits

should be adjusted accordingly. The start of each type of control chart is shown in Fig. 2. For the charts on the cells the central values for both cells and difference between cells should be based on the assigned values. The chart can either be kept on a run number or a time basis. The latter has the advantage that

one can estimate rate of drift if any cell shows a trend.
Table 2 Factors for Setting the 3 CT Control Limits for the Designs in Appendix t Number of Cells in Group

Ho. of cells Excluded from

3 4 5 6

1.0000

Cell Values

0.9186 1.2000 1.1260

1* 0.8660

1.500 0.8660

1.2247

1. 1619

1. 5000 1.1071

1.3512

2*

0.1.7320 0.750

1. 2990

1.0954 1.5492

1 . 0794 1.3839

3* ::: 1. 5000

• 9^87 1.6432

1.0000

1.4392

1.2247 Residual (P) 0.8660 0.9487 0.8018

s 1.7321

iccessive differences (vt - v.

,

n ]

1. 5000 1.8974 (1)

1.9^5 0.888

Standard deviations Upper limit Central line 1.585 0.950

1.737 0.933

1.552

0.963 cells included in the

Ls exc luded from the n (1) For differences (l-2)( 2-3) (4-5) ( 5-6) the limit i For differences (4-5)(6-l) the limit is 1.7525-

If a cell should "go bad" and be removed from the 11 limits should be altered accordingly (see Table 2). It control charts on the cells only indicate change in the

in, but still kept in the group, then the

; important to bear in mind that the

if of cells relative to each other. If the whole group is changing it will not show up in any of the charts and can only be ascertained by comparison with other cells whose values are known. This situation does occur because small groups

of cells are usually from the same manufacturer and lot, and therefore have similar aging c" i sties.

VI. Systematic Errors Ideally a measurement process should be free of systematic error, the case. In fact, the residual P is a systematic error. Its effect c readily removed and its magnitude estimated using the suggested designs are not so easily detected. Indications of their presence in some case

lis is not often

;he values of the cells is Other systematic errors

;an be obtained by analysis

of deviations from t 1 the £ '. group of cells.
— 0.7 |o.6

S 0.5 £

o 04

" 0.3 z 0.2 £ 0.1 5

Upper Limit = p + 3 ( 0" of eel value) — o _ o

M

= Accepted O ° o — o

Lower Llmit = p — 3 ( cr of eel

1 1 1 1 1 1 1

2.6 2.5 2.4 2.3

2 2

(V,-M)-(V2 -M) +3(<r of Difference) O _ o - ° (V,-M)-(V2 -M)

2.1 2.0

1.9

1.8 1.7

O O

(V|-M)-(V2 -M) -3 (a of Difference)

1 1 I 1 1 II

RUN NUMBER

2 3 4 5 6 7 RUN NUMBER

Control Chart for value of a cell. Control Chart for the difference between

0.12 - _ 0.10 —

a 0.08

<] 0.06

u." 0.04 sw 0.02 i

° 9 -0.02 a -0.04 -0.06

3(o- of Residual

)

o

o

o — 3(o- of Residual)

1 1 1 1 1 1 RUN NUMBER

~ 0.7 io.e

2 0.5 o 00.4 a 0.3

Constant x a o

Accepted Value of a S

U.2 §0.1

<

o

O

1 1 1 o

1

O

1 1 1 2 3 4 5 6 7 RUN NUMBER

Control Chart for standard deviation of s single observation. Control Chart for residual.

Figure 2 Typical control charts for maintaining surveillance c i group of standard cells.
Detection is based on the assumption that the deviations for a pai cell 2, cell 1 - cell 3, etc.) are independent in successive runs. If the magnitude and sign of corresponding deviations from success one would suspect the presence of a systematic error. Such an analysis groups graphically by plotting the deviations of one run as a function

ive runs tend to agree then

can be done on 5 and 6 cell

of the second run. If the deviations tend to fall o: would suspect a systematic errc Figure 3 shows an example with Figure 3 by adding 0.3 nV to tl such as failing to correct for

(l) cause the deviations to st: (3), introduce a bias into eacl particular set of observations.

straight line having slope 1 and passing through the origin, then c ror. If there is none then the points would be distributed random!

io systematic error present. Figure k was created from the data absolute value of each observation to simulate an offset error

he instrument zero. The presence of such a systematic error wi! ng along the line, (2) inflate the standard deviation, and calculated v. . The magnitude of the latter will depend on the Instead of the model for a single observation being

E(y ) = X - X + P

i of Figure 3, and as assumed in the appendi:

whei ; the s -o offset. ifitr

For sets with less than 5 cells one would examine the deviations of successive runs for patterns.

If the deviations for a given observation have the same sign and approximately the same magnitude,

one would suspect a possible systematic error. Studies are being conducted to develop relatively

simple tests that may be used to detect the presence of many types of systematic errors.

.ble c Failure to make applies Zero offset Operator reading bias Operator setting bias

Leakage

3 by r 3 complete, but n
0.4

0.3 * 02

o o y^

s 0.1 s°

'o hia

-0.1 o/o

z -02 /I

o

1 1 Mil

q: -0.3 £ -0.4 -

O /°

O /^

o X

O I I I I I I -0.3 -0.2 -0.1 O.i 0.2 0.3 0.4 RUN I DEVIATIONS ( pv) Figure 3 Youden plot for two runs without systematic

Figure k Youden plot for two runs with systematic error.

[1] Hamer, W. J., "The Volt Standard"

56 (1966), pp. 101-108.

[2] Harris, F. K., Electrical Mea:

3 To Gaithersburg, Maryland, J. Wash. Acad, of Sci .

;, J. Wiley & Sons (1952), pp. I68-I85.

[3] Hamer, W. J., Standard Cells —Their Construction, Maintenance, and Characteristics, NBS Monograph 84 (1965), pp. 8-10.

[h] Hamer, W. J., ibid , pp. 5-8.

[5] ASTM Manual on Quality Control of Materials, Special Technical Publication 15-C, Part 3, January 1951.

[6] Natrella, M. G. , Experimental Statistics, NBS Handbook 91 (1963), pp. 18-1, 18-4.

[7] Pontius, P. E. , Measurement Philosophy of the Pilot Program for Mass Measurement, NBS Tecl nical Note 288, May 1966.
Designs for Groups of 3, b, 5 and 6 Saturated Standard Cells
APPENDIX A

Designs for Groups of 3, k, 5 and 6 Saturated Standard Cells TABLE A-l

THE IWTERCOMPARISOTJ OF A GROUP OF THREE SATURATED STANDARD CELLS

Given: The emf s of three saturated standard cells (Vi, V2 and V3 ) are assigned by cali- brating them in terms of a known standard of electromotive force. The mean of the group

Example

1. From an MBS Report of Calibration : Vj = 1.0182571 vt = -1.2 11V v2 = 1.0182535 v3 = -4.8 V3 = 1.0182643 v3 = +6.0 Mean = I.OI82583 sum = and the difference from the r cell is v = (V -M)

Assuming that there is a small constant

emf P associated with the measuring process then the expected value of a single observation is

E(y.) =

For all possible values of j and k the following schedule of measurements is convenient and requires changing the

connections to only one cell at a time. Measure- Cell in UNK Cell in :

7i = +4 8 mv

73 = -6 6

ya = -10.6

V4 = -3 4

ys = +7 k

*See Fig. 1 for definition of posj

: 1/6 (+4.8 -6.6 -10.6 -3.4 +7.4 +10.4)

.333 UV
.-1 Continued

t. Estimation of v.

:

4. va = Vi-M = 1/6 ( yi + y2 -y4 -y5 ) V\ = 1/6 (4.8 -6.6 +3.4 -7.4) = -.967 nv

v2 = V2 -M = 1/6 (-yi +y3 +y4 -yB ) vs = 1/6 (-4.8 -10.6 -3.4 -io.it) = -4.867 v3 = V3 -M = 1/6 (-y2 -y3 +y5 +y6 ) v3 = 1/6 (+6.6 +10.6 +7.4 +10.4) = +5.833 Arithmetic Check = 2^ = (within round -off

)

Check = -.967 -4.867 +5.833 = -.001 nV

5. Calculation of y's, the predicted y's: 5- Yi = vj -v2 + P yi = ..967 +4.867 +.333 = +4.233 HV y2 = vt -v3 + P ys = -.967 -5.833 +.333 = -6.467 h = v2 -v3 + P ya = -4.867 -5.833 +.333 = -10.367

y4 = v2 -vT + p y* = -4.867 +.967 +.333 = -3.567 ys = v3 -v, + P ys = +5.833 +.967 +.333 = +7.133

ye = v3 -v2 + P ys = +5.833 +4.867 +.333 = +U.033

Arithmetic Check = Z y. = 6p Check = 2 y. = I.998

6. Calculation of the deviations (d. = y. -y.): 6. d i = yi -yi d l =4.8 -4.233 = .567 nv

d2 = y2 -ys d2 = -6.6 +6.467 = -.133

d3 = ya -y3 d3 = -10.6 +10.367 = -.233

d4 = y4 -y4 d4 = -3 A +3.567 = +.167

d5 = ys -ys d5 = 7-4 -7.133 = +.267

d6 = ys -ys ds = 10.4 -11.033 = --633

6 Check = 2 d. = (within round-off)

1=1 1 2d = .002

7. The standard deviation of a single observation (s) is

7. 6

s = 2d. 2 1=1 1 s=

v/^p =~. 55trt, where 3 represents the number of degrees of freedom in this error estimate.

8. Emf values of the cells 8. Mean (from Section l): 1.01825830

The emf's of the cells are calculated Vj = 1.01825830 - 0.00000097 = 1.01825733 by restoring the mean value to give v2 = 1.01825830 - 0.00000487 = 1.01825343

V = v. + M

v3 = 1.01825830 + 0.00000583 = 1.01826413
THE INTERCOMPARISON OF A GROUP OF 4 CELLS

Example

1. Given: The emfs of four saturated standard cells (Vi, V2 , V3 and V4 ) are assigned by- calibrating them in terms of a known stand1. From an NBS Report of Calibration: I.OI82459 vt = -4.1 UV ard of electromotive force. The mean of V2 = 1.0182488 v2 = -1.2 the group

4 M = l/4 2 V. i=l 1

v3 = 1.0182526 v3 = +2.6

1.0182527 v4 = +2.7

= 1.0182500 sum = and the difference from the mean of each

cell is

v. = (V.-M)

2. Assuming that there is a small constant emf 2. Observations: P associated with the measuring process, the

expected value of a single observation is

E(yi )= V -V

R

+ P jA; for j and k = 1,2,3,4- For all po ssible values of j and k the following schedule of measu rements is convenient, and in most cases requires changing

the connections to only one cell at a time. Measure- Cell in UNK Cell in REF

ment position position

yi 1 2 yi = -3.1 MV

y2 1 3 ys = -6.9

ys 2 3 y3 = -3.8

y4 2 4 y4 = -4.0

y5 3 4 ys = - .4 ys 3 1 ye = 6.3

y? 3 2 y? = 3.3

ye 4 2 y8 = 3.4

y9 4 1 y9 = 6.4 yu 4 3 yw = - .2 yn 2 1 yn = 2.7

yis 1 4 yis= -7.0

3. Estimation of P: 123. P = 1/12 Z y p = 1/12 (-3.1 -6.9 -3.8 -4.o -.4 +6. +3.3

1=1

p = +3.4 +6.4 -.2 +2.7 =7.0)

.275 nv
TABLE A-2 Contii

Vj = Vt -M = 1/8 (yt +y2 -ys -yg-yil +y*3 ) v2 = %-H = 1/8 (-yi +y3 +y4 -y, -ye +yn)

v3 = V3 -M = 1/8 (-y2 -y3 +y5 +y6 +y7

-310

)

*

4 = V4 -M = 1/8 (-y4 -y5 +y8 +y9 +y10 -yl3 ) Arithmetic Check = Z v. =0 (within

1 round -off)

vj = 1/8 (-3.1 -6.9 -6.3 -6.4 -2.7 -7.0) = -4.05 MV vs = 1/8 (3.1 -3.8 -4.0 -3.3 -3.4 +2.7) = -1.088

v4 = 1/8 (4.0 +.4 +3.4 +6.4 -.2 +7.0) = 2.625 -4.050 -1.087 +2.512 +2.625 = 0.001

1 of y's, the predicted y'E

yio = v* -V, + p yn = v2 -Vj + e

yis = Vl -V4 + p

Arithmeti : Check = 2

= -4.050 +1.088 -.275 = -3.237 uv

= -4.050 -2.512 -.275 = -6.837

= -1.088 -2.512 -.275 = -3.875

= -1.088 -2.625 -.275 = -3.988

'= 2.512 -2A 625 -.275 = -O.388

= 2.512 +4.050 -.275 = 6.287

= 2.512 +1.088 -.275 = 3.325

= 2.625 +1.088 -.275 = 3.438

= 2.625 +4.050 -.275 = 6.4oo

= 2.625 -2.512 -.275 = -0.162

, = -1.088 +4.050 -.275 = 2.687

3 = -4.050 -2.625 -.275 = -6.950

a = (-3.237 -6.837 -3.875 -3.988 -0.388 +6.287 +3.325 +3.^38 +6.4oo -0.162 +2.687 -6.950) = -3.300 Calculation of the deviatic L (d

t = y.-yj):

Arithmetic Cheek = Z d.

-3.1 +3.237 = .137 MV

-6.9 +6.837 = -.063 -3.8 +3.875 = .075 -4.o +3.988 = -.012

-0.4 + .388 = -.012

6.3 -6.287 = .013 3.3 -3.325 = -.025 3.4 -3.438 = -.037 6.4 -6.400 = -.2 +.162 = -.038

2.7 -2.687 = .013 -7.0 +6.950 = -.050

-.063 +.075 -.012 -.'012 +.013 - 02s -.038 +0 -.038 +.013 -.050) = The standard deviatioi

Z (d ) 2

Emf values of the cells Mean (from Section l): 1.01825000

V,, = 1.01825000 - 0. ooooo4o5 = 1.01824595 % = 1.01825000 - 0.00000109 = 1.01824891 va = 1.01825000 + 0.00000251 = 1.01825251

v4 = 1.01825000 + 0.00000262 = 1.01825262
TABLE A-3 THE INTERCOMPARISON OF A GROUP OF 5 CELLS Example

1. Given: The emfs of five saturated standard cells (Vj,, V2 , V3 , V4 and Vs ) are assigned

by calibrating them in terms of a known

standard of electromotive force. The mean

of the group is

k M = 1/5 2 V.

1. From an KBS Report of Calibration:

Vi = 1.0182538 Vj. = .8 nV

Vs = 1.0182531 v2 = .1 Vs = 1.0182518 v3 = -1.2

V4 = 1.0182532 v4 = .2 Vs = 1.0182531 vE = .1 and the difference from the meai cell is "- Mean = 1.0182530 sum =

v. = (V.-M)

2. Assuming that there is a small constant emf P associated with the measuring process then the expected value of a single2. Observation^.

E(y

j_)= V.-Vk

+ P j^kf for j and k= 1,2... 5 For certain values of j and k a set of 10 measurements having "left-right" balance

that is convenient and requires changing only one cell at a time is

1/10 (.5 +1.6 +.9 -.4 -1.5 -1.3 h -.8 -1.0 -.2)

1 of \

= Yi-M = 1/5 (yi +y2 -y8 -y9 )

= vb -m = 1/5 (-yi +y3 +y4 -yio)

. v3 -m = 1/5 (-ya -y3 +y5 +y6 )

= v4 -m = 1/5 (-y4 -y5 +y7 +y8 )

= v5 -m = 1/5 (-ys -y7 +y9 +y10 ) Arithmetic Check = 5 V=0 (within

L round -off)

= 1/5 (.5 +1.6 +.8 +1.0) = .78 11V

-- 1/5 (-.5 +.9 -- 1* +-2) = .04

. 1/5 (-1.6 -.9 -1.5 -1.3) = -1-06

= 1/5 (+.4 +1.5 +0. -.8) = .22

= 1/5 (1.3 +0. -1.0 -.2) = 0.02

5

:k: 2 v. = .78 +.04 -1.06 +.22 +.02=
TABLE A-3 Continued

5. Calculation of y's, the predicted y's: 5. yi = yi = Vj -v3 + P 78 -.04 -.22 = .52 nV h = vi "Va + P y3 = 78 +1.06 -.22 = 1.62

y3 = y

2 -v3 + P y3 = 04 +1.06 -.22 = .88 h = v3 -v4 + P y* = 04 -.22 -.22 = -.4

ys = v3 -y4 + P y5 = 1.06 -.22 -.22 = -1.50

ys = v3 -v5 + P ye = 1.06 -.02 -.22 = -1.30

y? = v4 -v5 + P y7 = 22 -.02 -.22 = -.02 ia = V4 -Vj + P y8 = 22 -.78 -.22 = -.78

ye = V5 -V! + P yB = 02 -.78 -.22 = -.98 yw = vB -v3 + P Ao = 02 -.04 -.22 = -.24 Arithmetic Check: Z y. = 10P Check Z y = -2.20 = 10(-.22)= -2.20

6. Calculation of the deviations (d. = y.-y.): 6. di = yj -yi di = 5 -.52 = -.02 juV d2 = ys -ys d3 = 1.6 -1.62 = -.02

d3 = ys -ys d3 = 9 -.88 = .02

d4 = y4 -y4 d4 = .k + .4 =

dE = yE -ys d5 = 1.5 +1.50 =

ds = y6 -ys ds = -1.3 +1.30 =

d7 = yv -y-7 d7 = 0. +.02 = .02

d8 = y8 -y8 de = .8 +.78 = -.02

d9 = y9 -y9 d9 = -1.0 +.98 = -.02

a.10 = yio -yio 10

dio = -.2 +.2k = .Ok Check Z d. = (within round-off) Check = (-.02 -.02 +.02 +.3(0) +.02 -.02 1-1 1 -.02 +.0k) =

7. The standard deviation of a single 7. observation (s) isrio

/ z a.

\ / i-l x

s = /^=.028,V

V 5 where 5 represents the number of degrees of freedo n in this error estimate.

8. Emf values of the cells 8. Mean from Section l): 1.01825300 The emf's of the cells are calculated by Vj = L. 01825300 + 0.00000078 = 1.01825378

restor mg the mean value to give % = 1.01825300 + 0.00000004 = 1.01825304

Ys = L. 01825300 - 0.00000106 = 1.0182519^

V. = v. + M v4 = 1.01825300 + 0.00000022 = 1.01825322

v5 = 1.01825300 + 0.00000002 = 1.01825302
THE TNTERCOMPARISON OF A GROUP OF 6 CELLS Example

1. Given: The emfs of six saturated standard 1. From an WBS Report of Calibration: cells (Vj. Vs, V3 , V4 , Vs and Ve ) are assigned by calibrating them in terms of a vi = 1.0182605 Vj = 10.52 known standard of electromotive force. The V2: = 1.0182655 v3 = 15.52 mean of the group is V3 = 1.0182U66 v3 = -3.38

6 M = 1/6 I V. i=l 1 V4 = 1.0l82it76 v4 = -2.38

V5 = 1. 0182416 v5 = -8.38

Vs = 1.0182381 v6 = -11.88

and the difference from the mean of each Mean = 1.01824998 sum = cell is V

± = (V

i -M)

2. Assuming that there is a small constant emf 2. Observations: P associated with the measuring process, the expected value of a single observation

E(y.)= v V +P j&; for j and k = 1,2. ..6 For certain values of j and k a set of 15 measurements, 12 of which are ' left -right" balanced, can be made. Measure- Cell in UNK Cell in KEF

ment position position

1 1 2 yi = -5.4 nV

2 1 3 ys = 13.7

3 2 3 y3 = 18.8

It 2 k Y4, = 17.7

5 3 h ys = -1.3

6 3 5 y6 = 4.8

7 k 5 y7 = 5.9 8 k 6 ys = 9.5

9 5 6 ys = 3.5

10 5 1 yio = -19.1

11 6 1 yn = -22.7 12 6 2 yia= -27.9

13 1 4 yia= 12.5

l4 2 5 yi4= 23.7

15 3 6 yis= 8.4

3. Create a »et of sums Q., S and T: 3. Qi = yi +ys -yio -yi, +yi3) Qi = (-5.4 +13.7 +19.1 +22.7 +12.5)= +62.6

Qs = -yi +y3 +y4 -yis +yi4 ) Qs = (5.4 +18.8 +17-7 +27.9 +23.7)= +93-5

Q3 = -ya -ya +ys +ys +y»s] Q3 = (-13.7 -18.8 -1.3 +4.8 +8.4)= -20.6

04 = -y* -ys +y? +ya -yi3) Q4 = (-17.7 +1.3 +5-9 +9-5 -12.5)= -13.5

Qs = -ye -7-7 +y9 +yio -yi 4 ) Q5 = (.4.8 -5.9 +3-5 -19.1 -23.7)= -50.0

Qe = -ys -ys +yn +yi2 -y» 5) Qs = (-9-5 -3.5 -22.7 -27.9 -8.4)= -72.0

S = ii +Qs +Qa s = (62.6 +93.5 -20.6)= 135.5 T = sum of all measurement

6

s T = (-5.4 +13.7 +18.8 +17.7 -1.3 +4.8 +5.9

+9-5 +3.5 -19.1 -22.7 -27.9 +12.5 +23.7 +8.4) = 42.1 Arithmeti Check = Z Q. = Check = (62.6 +93-5 -20.6 -13.5 -50.0 i=l 1 -72.0) =

h. Calculate P: k. p_ 3^8 P = 126 -\- W5.5.. .. 219 MV
TABLE A-4 Contii

5. Calculate v's: 5- vj = 1/6 (Qi-P) vt = i/6 (62.6 +.219) = 10.470 mv

vs = 1/6 (Qa-f) vs = 1/6 (93.5 +.219) = 15.620

v3 = 1/6 (Q3 -P) v3 = 1/6 (-20.6 +.219) = -3-397 v4 = 1/6 (Q^ +P) v4 = i/6 (-13.5 -.219) = -2.286 v5 = 1/6 (Q5+P) vs = 1/6 (-50.0 -.219) = -8.370

vs = 1/6 (Qs+f)

6 Check = Z v. =

vs = 1/6 (-72.0 -.219) = -12.036 Check = 1/6 (62.819 +93.719 -20.819

i=l 1 -13.781 -50.219 -72.219) =

6. Calculation of y's, tt ctedy-s: 6. Yi = vj -vs + P yj = 10.^70 -15.620 -.219 = -5.369

ys = Vi -v3 + P y2 = IO.V70 +3.397 -.219 = 13.648 y3 = v2 -v3 + P y3 = 15.620 +3-397 -.219 = 18.798

y4 = v2 -v4 + p y4 = 15.620 +2.286 -.219 - 17.687 y5 = v3 -v4 + P y5 = -3-397 +2.286 -.219 = -1.330

y6 = y3 -V5 + P y6 = -3-397 +8.370 -.219 = 4.754 y7 = y4 -v5 + P y7 = -2.286 +8.370 -.219 = 5.865 ye = y4 -y6 + P y8 = -2.286 +12.036 -.219 = 9.531

yB = y5 -y6 + P ys = -8.370 +12.036 -.219 = 3.447 y10 = v5 -Vj + P yio = -8.370 -10.470 -.219 = -19.059

y11= y6 -VX + P yn= -12.036 -10.470 -.219 = -22.725 yi2= V8 -Vg + P yis= -12.036 -15.620 -.219 = -27.875

y13 = y\ -y4 + P yl3 = 10.470 +2.286 -.219 = 12.537

yi 4 = vs -vs + P yi4 = 15.620 +8.370 -.219 = 23.771 y15 = v3 -v6 + P yl5 = -3-397 +12.036 -.219 = 8.420

7. Calculation of the deviations (d

i = y±-$i>- 7. di = yi -y-L di = -5.4 +5.369 = -.031 MV

da = y3 -yE d3 = 13.7 -13.648 = .052

d3 = y3 -y3 d3 = 18.8 -18.798 = .002

d4 = y4 -y4 d4 = 17.7 -17.687 = .013

d5 = ys -y5 d5 = -1.3 +1.330 = .030

ds = ys -ye ds = 4.8 -4.754 = .046

d7 = y7 -y? d7 = 5.9 -5.865 = .035

de = yB -ye de =9-5 -9.531 = -.031

d9 = y9 -y9 d9 = 3.5 -3-447 = .053

dio = yio -Ao d10 = -19.1 +19.059 = -.041 d,i= yu -fn dlT = -22.7 +22.725 = .025 dis= yl2 -y13 dl2 = -27.9 +27.875 = -.025

dl3 = y l3 -yi3 d 13 = 12.5 -12.537 = -.037 di4= y14 -yi4 d i4 = 23.7 -23.771 = -.071

d l5 = y15 -yt s d 15 = 8.4 -8.420 = -.020

8. The standard deviation

vation (s) is of a single obser- 8.

, /z(di)2 V

9

s =p^2° = .0490 ,V where 9 represents the degree s of freedom

for error.

9. Emf values of the cells 9. Mean (from Section l): 1.01824998

The emf's of the celis' are calculated by v, = 1.01824998 + 0.00001047 = 1.01826045 restoring the mean value to give V2 = 1.01824998 + 0.00001562 = I.OI826560

v 3 = 1.01824998 - 0.00000340 = 1.01824658

V. = v + M v4 = 1.01824998 - 0.00000229 = 1.01824769

v5 = 1.01824998 - 0.00000837 = I.0l824i6i

v6 = 1.01824998 - 0.00001204 = 1.01823794

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